High Precision Optical Characterization of Carrier Transport Properties in Semiconductors

ABSTRACT

A precise optical technique for measuring electronic transport properties in semiconductors is disclosed. Using tightly focused laser beams in a photo-modulated reflectance system, the modulated reflectance signal is measured as a function of the longitudinal (Z) displacement of the sample from focus. The modulated component of the reflected probe beam is a Gaussian beam with its profile determined by the focal parameters and the complex diffusion length. The reflected probe beam is collected and input to the detector, thereby integrating over the radial profile of the beam. This results in a simple analytic expression for the Z dependence of the signal in terms of the complex diffusion length. Best fit values for the diffusion length and recombination lifetime are obtained via a nonlinear regression analysis. The output diffusion lengths and recombination lifetimes and their estimated uncertainties may then be used to evaluate various transport properties and their associated uncertainties.

CROSS-REFERENCE TO RELATED PATENT APPLICATIONS

The present invention claims priority to U.S. Provisional PatentApplication Ser. No. 62/498,685, filed on Jan. 5, 2017, entitled “HighPrecision Optical Characterization of Carrier Diffusion Length, CarrierMobility, and Sheet Resistance,” which provisional patent application ishereby incorporated herein by reference in its entirety for allpurposes.

BACKGROUND Field of the Invention

The present invention relates to optical measurement of electronictransport properties in semiconductors and, more particularly, to highprecision measurement of carrier diffusion lengths, recombinationlifetimes, diffusion coefficients, mobilities, and related carriertransport properties using photo-modulated reflectance techniques.

Background of the Invention

Semiconductor device performance depends fundamentally on the transportproperties of electronic charge carriers within the semiconductor. Inmodern device manufacturing, many process steps have the potential toalter or degrade carrier transport properties. Spatial and temporalshifts in process conditions cause variations in carrier transportproperties, resulting in device timing variability and loss ofmanufacturing yield. This systematic type of variability ispreferentially addressed through the application of “in-line” processcontrol techniques, which require measurements directly on productwafers as they move through the manufacturing flow (i.e. betweenprocessing steps). Thus in-line process control is enabled by theability to rapidly and non-destructively measure properties indicativeof final device performance with micrometer scale spatial resolution.

Carrier transport properties are conventionally measured usingfour-point probe (4PP) and/or Hall effect techniques. Although thesetechniques have the requisite measurement capability, they are notsuitable for in-line process control due to their destructive and timeconsuming nature. At present, even the most advanced “micro-4PP” and“micro-Hall” techniques require contact with the sample. Such contact isdisfavored in process control applications due to the attendantpossibility of particulate contamination.

Optical techniques are ideal for process control of semiconductormanufacturing as they are non-contact, can be performed rapidly, andprovide micrometer scale resolution. For example, ellipsometry has beenwidely adopted in the semiconductor industry for the non-contactmeasurement of film thicknesses. Furthermore, numerous opticaltechniques have been developed for measurement of carrier transportproperties. Surface photo-voltage (SPV), photo-conductance decay, freecarrier absorption, photo-luminescence, and time resolved THzspectroscopy are among the techniques frequently reported in the priorart. However, in each case some particular feature of the techniquepresents a significant impediment to use for in-line process control.For example, the capacitive probes used in SPV techniques suffer fromlimited sensitivity (e.g. from stray capacitances) and poor spatialresolution. The spatial resolution of SPV limits its application to“at-line” process monitoring using unpatterned blanket wafers. Anotheroptical technique, THz spectroscopy, is sensitive to carrier mobilitybut requires a complicated (and expensive) experimental setup. Thus atpresent there remains an outstanding need to rapidly andnon-destructively measure carrier transport properties betweenprocessing steps.

The techniques disclosed herein generally utilize photo-modulatedreflectance (PMR) techniques to characterize transport properties ofelectronic charge carriers. PMR techniques have been particularlyimportant in basic research on semiconductors [F. H. Pollak, “ModulationSpectroscopy of Semiconductors and Semiconductor Microstructures,” inHandbook on Semiconductors, Vol. 2, edited by M. Balkanski, pp. 527-635(North-Holland, Amsterdam, 1994).]. The conventional PMR setup consistsof an intensity modulated pump light beam used to modulate thereflectivity of the sample, a second continuous wave light beam used toprobe the reflectance of the sample, an optical system for directing thepump and probe light beams to the sample, and for directing thereflected probe light onto a photodetector, and a signal processor tomeasure the differential reflectance. In general, the pump beam is usedto modulate the free charge density in a semiconductor sample (i.e. viaphoto-injection), and hence modulate one or more physical quantities(such as, for example, near surface electric fields), thereby inducing atime periodic variation in the reflectivity of the sample. The pumplight is typically modulated at a known frequency so that a phase-lockeddetection circuit may be used to suppress unwanted noise, resulting inthe ability to detect reflectance changes at the ppm level. It may beappreciated that the pump beam in a PMR setup is analogous to the pumpbeam in a SPV or junction photo-voltage (JPV) measurement. The measuredPMR signal is the relative change in the reflected probe light intensityas the intensity modulated pump radiation interacts with the sample andconsists of a vector characterized by an amplitude and a phase. Thenormalized amplitude is the induced change in reflectance (AC) dividedby the linear reflectance (DC), whereas the phase is the relative lag ofthe reflectance change with respect to the driving phase. This phase lagis due to the non-instantaneous relaxation dynamics of the chargecarriers within the sample.

For probe beam wavelengths that are resonant with semiconductorinterband transitions, the PMR signal arises from an electro-modulationeffect (i.e. the Franz-Keldysh effect). This provides a means foroptical detection of semiconductor bandstructure and internal electricfields through the appearance of sharp, third derivative spectra [D. E.Aspnes, “Linearized Third-Derivative Spectroscopy with Depletion-BarrierModulation,” Phys. Rev. Lett. 28, 913-916 (1972); D. E. Aspnes and J. E.Rowe, “Resonant Nonlinear Susceptibility: Electroreflectance in theLow-Field Limit,” Phys. Rev. B 5, 4022-4030 (1972).]. Moreover, inaccord with the Poisson relation, the experimentally measured PMR signalunder such conditions is:

ΔR/R=2qN _(e)∧(λ)/ε_(s) ×ΔV,  (1)

where q is the electronic charge, N_(e) is the active dopantconcentration, ∧(λ) is a line-shape function determined by thesemiconductor bandstructure (λ is the probe beam wavelength), ε_(s) isthe static dielectric constant, and ΔV is the pump induced photo-voltage[N. Bottka et al., “Modulation Spectroscopy as a Tool for ElectronicMaterial Characterization,” J. Electron. Mater. 17, 161-170 (1988).].Eq. (1) is valid for depleted surfaces provided the electric field isnot too inhomogeneous [D. E. Aspnes and A. Frova, “Influence ofSpatially Dependent Perturbations on Modulated Reflectance andAbsorption of Solids,” Solid State Comm. 7, 155-159 (1969). (“Aspnes1969”)].

In conventional PMR analysis, a nonlinear regression analysis is used toadjust the variables within a parameterized third derivative functionalform to provide an optimum fit to the PMR signal as a function ofwavelength. The nonlinear regression analysis outputs the best-fitparameters, a statistical estimate of the error in the outputparameters, and an overall “goodness-of-fit” measure. The errorestimates generally depend upon (i) the measurement uncertainties ateach data point, (ii) the number and spacing of the data points, and(iii) the match of the analytic model to the data [W. H. Press et al.,“Modeling of Data,” in Numerical Recipes in Fortran 77: The Art ofScientific Computing, 2nd ed., pp. 650-700 (Cambridge University Press,New York, 1996).]. The conventional tit procedure is typically used todetermine semiconductor interband transition energies, widths, and/oramplitudes. As noted, the third derivative form is large only nearbystrong optical absorptions in the semiconductor band structure, and thusmay be used to measure interband transition energies with greatprecision. Spectroscopic PMR techniques are also useful to measurematerial composition and/or physical strain in semiconductors sinceinterband transition energies shift under composition and strain. Thesefeatures account for the historical importance of PMR in basic researchon semiconductors.

More recently, PMR techniques using a monochromatic laser probe beamsuitable for in-line process control applications have been developed[W. W. Chism, “Method of Photo-Reflectance Characterization of Strainand Active Dopant in Semiconductor Structures,” U.S. Pat. No. 7,391,507,issued Jun. 24, 2008.]. For example, provided the probe beam wavelengthis suitably chosen, the PMR signal is linearly proportional to theactive dopant concentration, which is particularly convenient forcontrol of advanced annealing processes in modern semiconductormanufacturing [W. Chism et al., “Photo-Reflectance Characterization ofUSJ Activation in Millisecond Annealing,” J. Vac. Sci. Technol. B 28,C1C15-C1C20 (2010). (“Chism 2010”)]. Moreover, since the photo-injectedcarriers will always reduce the latent voltage, the PMR signal reversessign under a reversal of the surface polarity. Thus the sign of PMRsignal may be used to infer which type of carrier is present. “Z-scan”PMR techniques, wherein the position of the sample surface is scannedthrough focus while spatial distortion of the reflected probe beam isanalyzed via an aperture in the far field, have also been used todistinguish the refractive and absorptive components of the PMR signal[W. W. Chism, “Method and Apparatus of Z-scan PhotoreflectanceCharacterization,” U.S. Pat. No. 8,300,227, issued Oct. 30, 2012.]. Incertain situations, the coherent wavefront of the laser probe beam maybe used to isolate the refractive component of the PMR signal, greatlysimplifying the data analysis [W. Chism and J. Cartwright, “LaserPhoto-Reflectance Characterization of Resonant NonlinearElectro-Refraction in Thin Semiconductor Films,” Thin Solid Films 520,6521-6524 (2012).].

Despite these advances, PMR is not widely used to determine electronictransport properties in semiconductors. Rather, SPV and otheraforementioned optical techniques are more commonly used for non-contactmeasurement of carrier transport properties. As noted, the pump beam ina PMR setup is analogous to the pump beam in a SPV or JPV measurement.Also, in accord with Eq. (1), the PMR signal is linearly proportional tothe SPV (ΔV). The SPV depends on the pump beam and the physics of itsinteraction with the sample [L. Kronik and Y. Shapira, “Surfacephotovoltage phenomena: theory, experiment, and applications,” Surf.Sci. Rep. 37, 1-206 (1999).]. Thus, determination of carrier transportproperties from SPV measurements generally requires prior knowledge ofthe sample physics, e.g. pump absorption depth, surface and/or interfacerecombination velocities, etc. Moreover, only in certain limiting casesare the desired transport properties directly related to the measuredquantities (SPV amplitude and phase). However, the SPV amplitude isgenerally linear in the pump intensity provided the photo-injection issmall with respect to the restoring current [D. K. Schroder, “CarrierLifetimes in Silicon,” IEEE Trans. Electron Devices 44, 160-170 (1997);J. E. Park et al., “Silicon epitaxial layer lifetime characterization,”J. Electrochem. Soc. 148, G411G416 (2001). (“Park 2001”)]. In this casethe SPV will exhibit the spatial dependence of the excess carrierdensity. In the one-dimensional (1D) limit, the SPV may be obtained fromthe solution of the 1D differential equation for the modulated thecarrier density [D. K. Schroder, “Surface voltage and surfacephotovoltage: history, theory and applications,” Meas. Sci. Technol. 12,R16-R31 (2001).]. In the three-dimensional (3D) limit, the excesscarrier density with a Gaussian laser source involves a Gaussianweighted zero-order Hankel transform of the 1D solution and thereforemust be treated numerically [J. Opsal et al., “Thermal-wave detectionand thin-film thickness measurements with laser beam deflection,” Appl.Opt. 22, 3169-3176 (1983).]. Thus in view of the appearance of the SPVin the PMR signal, the difficulties encountered in the use of SPV todetermine carrier transport properties are likewise present inconventional PMR techniques.

However, as disclosed herein, use of laser beams for both the pump andprobe in a PMR setup means the reflected probe beam may be treateddirectly according to the analytically tractable method of Gaussiandecomposition [D. Weaire et al., “Effect of low-power nonlinearrefraction on laser-beam propagation in Insb,” Opt. Lett. 4, 331-333(1979); M. Sheik-Bahae et al., “High-sensitivity, single beam n₂measurements,” Optics Lett. 14, 955-957 (1989); M. Sheik-Bahae et al.,“Sensitive Measurement of Optical Nonlinearities Using a Single Beam,”IEEE J. Quant. Electron. 26, 760-769 (1990); D. V. Petrov et al.,“Reflection Z-scan technique for measurements of optical properties ofsurfaces,” Appl. Phys. Lett. 65, 1067-1069 (1994).]. The numericaltreatment required by the prior art in the 3D limit is thereby avoided.Moreover, as demonstrated herein, the analytic expression resulting fromtreatment by the method of Gaussian decomposition enables a nonlinearregression analysis to directly determine carrier transport propertieswithout prior knowledge of the sample physics.

SUMMARY OF THE INVENTION

The present invention relates to methods for determining electronictransport properties in a semiconductor sample. More particularly, thepresent invention provides methods for the direct, high precisionoptical characterization of carrier diffusion lengths, recombinationlifetimes, and/or diffusion coefficients using Z-scanning laserphoto-modulated reflectance techniques. Once these carrier propertiesare measured, various additional electronic transport propertiesincluding the carrier mobility, the carrier effective mass, and thesheet resistance may the evaluated with high precision. Thus highprecision optical characterization of carrier transport properties isachieved.

The methods disclosed herein generally utilize laser beams for both thepump and probe beams in a photo-modulated reflectance apparatus. Thepump beam is an intensity modulated laser beam used to modulate thecharge density in a semiconductor sample. The pump beam induces amodulated excess carrier density within the sample within a radius ofmodulation ω_(m)=(ω_(p) ²+L_(d) ²)^(1/2), where ω_(p) is the pump beamradius and L_(d) is the carrier diffusion length. In low injection, theinduced photovoltage will exhibit the spatial dependence of the excesscarrier density. Thus the spatial profile of the induced photovoltagedepends on the carrier diffusion length. A second continuous wave laserbeam is used to probe the change in reflectance of the sample as theintensity modulated pump radiation interacts with the sample. PMRsignals are acquired as a function of the distance of the sample surfacefrom the focal plane of the pump beam. As the sample is stepped throughfocus, the area of modulation varies according to the relation ω_(m)²(Z)=ω_(p) ²(Z)+L_(d) ², where Z is the distance from the pump beamwaist to the sample surface and ω_(p)(Z) is the radius of the incidentpump beam at the sample surface (i.e. ω_(p) ²(Z)=ω_(p) ²[1+(Z/z_(p))²],where z_(p)=πω_(p) ²/λ_(p) is the Rayleigh range of the pump beam andλ_(p) is the pump beam wavelength). The modulated component of thereflected probe beam is also a Gaussian beam with radius defined by theincident probe beam radius and the radius of modulation. The reflectedprobe beam is collected and input to the detector, thereby integratingover the transverse profile of the reflected probe beam. This spatialintegration results in a simple analytic expression for the Z dependenceof the PMR signal which depends solely on the focal parameters and thecomplex carrier diffusion length. The analytic expression enables anonlinear regression analysis which outputs the carrier diffusionlength, diffusion coefficient, and/or recombination lifetime, includingerror estimates on the output parameters, and a “goodness-of-fit”measure. Thus the numerical treatment required by the prior art isavoided by treating the reflected laser probe beam directly by themethod of Gaussian decomposition. Moreover, as the complex diffusionlength is the only sample parameter on which the Z profile of the PMRsignal depends, the techniques disclosed do not require knowledge orinterpretation of the physics underlying the PMR signal at Z=0.

A regressive fit to the PMR data is performed using a parameterizedfunction wherein at least one carrier transport property is treated as avariable parameter. The error estimates on the output fit parametersdepend upon the measurement uncertainties at each data point, the numberand spacing of the data points, and the match of the analytic model tothe data. The PMR measurement uncertainties approach the ppm levelwhereas the analytic model presented herein is broadly applicable. Thusin practice, the technique only depends upon the number and spacing ofthe data points in Z. In short, the analytic parameterization for the Zdependence of the PMR signal in terms of the complex diffusion lengthenables the direct, high precision determination of carrier diffusionlengths and/or recombination lifetimes using data obtained from a simpleoptical arrangement.

Once the diffusion length and recombination lifetime (and theirestimated uncertainties) are known from the fit procedure, these carriertransport properties may be used to evaluate the diffusion coefficient,or equivalently, the carrier mobility (via the Einstein relation).Alternatively, low and high frequency PMR amplitude curves can be fit toobtain the carrier diffusion length and diffusion coefficient,respectively (which then may be used to evaluate the recombinationlifetime). And once the mobility and the recombination time are known,the carrier effective mass may be evaluated. Similarly, once themobility is known, it may be used in combination with the active dopantconcentration in order to determine the sheet resistance (orequivalently, the sheet conductance). Thus the present disclosuredescribes the of Z-scanning laser PMR techniques to rapidly,nondestructively and precisely characterize electronic transportproperties including carrier diffusion lengths, recombination lifetimes,diffusion coefficients, mobilities, effective masses, and sheetresistances.

Furthermore, the Z-scanning laser PMR techniques disclosed here arefully applicable to PMR techniques which directly detect the excesscarrier density (i.e. via the Drude effect). The experimentally measuredPMR signal under such conditions is:

ΔR/R=q ²λ²/2π²ε_(s) n(n ²−1)mc ² ×ΔN,  (2)

where n is index of refraction, m is the carrier effective mass, and ANis the pump induced excess carrier density [R. E. Wagner and A.Mandelis, “A Generalized Calculation of the Temperature and DrudePhoto-Modulated Optical Reflectance Coefficients in Semiconductors,” J.Phys. Chem. Solids 52, 1061-1070 (1991).]. This Drude effect PMR signalis proportional to the square of the probe wavelength. Thus directmeasurement of carrier modulation is preferred at longer wavelengths,such as, for example, in the near-IR. Such wavelengths generallypenetrate far more deeply into the bulk and therefore do not provide thenear surface specificity available when the wavelength of the probe isselected to coincide with interband transitions in the UV (i.e. suchthat the PMR signal arises from the Franz-Keldysh effect). Nevertheless,the AN factor present in the expression for the Drude effect PMR signalwill exhibit the same spatial profile dependence on diffusion length asexhibited by the photo-voltage. In particular, the pump will induce anexcess carrier density within the same radius of modulation as exhibitedby ΔV. Thus the inventive techniques disclosed herein are fullyapplicable to PMR probe beam wavelengths selected to detect the excesscarrier density via the Drude effect.

There has thus been outlined, rather broadly, the more importantfeatures of the invention in order that the detailed description thereofmay be better understood, and in order that the present contribution tothe art may be better appreciated. There are additional features of theinvention that will be described hereinafter.

In this respect, before explaining at least one embodiment of theinvention in detail, it is to be understood that the invention is notlimited in its application to the details of construction and to thearrangements of the components set forth in the following description orillustrated in the drawings. The invention is capable of otherembodiments and of being practiced and carried out in various ways.Also, it is to be understood that the phraseology and terminologyemployed herein are for the purpose of the description and should not beregarded as limiting.

BRIEF DESCRIPTION OF THE DRAWINGS

The following drawings form part of the present specification and areincluded to further demonstrate certain aspects of the presentdisclosure. The disclosure may be better understood by reference to oneor more of these drawings in combination with the detailed descriptionof specific embodiments presented herein.

FIG. 1 shows an exemplary schematic laser PMR arrangement that may beused to perform Z-scanning laser PMR measurements in accordance with thepresent invention.

FIG. 2 shows an exemplary focal geometry of the pump and probe (bothincident and reflected) beams at Z=0. Exemplary profiles of thereflected AC probe beam for different values of L_(d) are also shown.

FIG. 3 shows Z-scanning laser PMR amplitude data and fits illustratingthe effect of near surface damage on carrier diffusion length in shallowelectrical junctions formed in silicon. The more narrow Z profileindicates a shorter diffusion length.

FIG. 4 shows Z-scanning laser PMR phase data and fits generated from thesame pair of samples as shown in FIG. 3, illustrating the effect of nearsurface damage on carrier recombination lifetime in shallow junctions.The broader phase data corresponds to the more sharply peaked amplitudedata, i.e. the broader phase profile indicates a shorter recombinationlifetime.

DETAILED DESCRIPTION

The following discusses use of the method of Z-scanning laserphoto-modulated reflectance characterization of electronic transportproperties in semiconductors. It is understood that the method of thepresent description may be used to determine electronic transportproperties in any semiconductor, the discussion of exemplary siliconsemiconductors considered to be exemplary only and in no way limiting inscope. It should be appreciated that the present invention providesnumerous applicable inventive concepts that may be embodied in a varietyof specific contexts. The specific embodiments discussed herein aremerely illustrative of specific ways to make and/or use the inventionand are not intended to delimit the scope of the invention.

The present invention is based upon profiling of the output signals of alaser PMR system as the sample is stepped through focus. The specificembodiments discussed here use laser beams having a cylindricallysymmetric Gaussian intensity distribution for both the pump and probebeams in a PMR apparatus. However, it should be appreciated thetechnique of the present disclosure may be accomplished with ellipticalGaussian profile beams or with any other beam profile which admits ananalytic parameterization for the Z dependence of the PMR signal interms of an electronic transport property of the sample. Likewise, itshould be appreciated the technique of the present disclosure may beaccomplished using a variety of focal geometries, including, forexample, focusing the pump beam onto the sample at normal incidencewhile independently focusing the probe beam onto the sample with anoblique angle of incidence. As shown in FIG. 1, a schematic arrangement101 that may be used to perform Z-scanning laser PMR measurements inaccordance with the present invention comprises a lock-in amplifier 102,a pump laser 103, a probe laser 104, an optical system, which maycomprise a dichroic beamsplitter 105, a microscope objective 106, aprobe telescope 107, a pump telescope 108, a reflecting semiconductorsample 109, a polarization beamsplitter 110, a quarter wave plate 111, acolor filter 112, a dielectric mirror 113, a collection lens 114, aphotoreceiver 115, and a computer (or other controller) 116.

In an exemplary embodiment of the present invention, the pump laser 103intensity is directly modulated via reference signal from the lock-inamplifier 102. The pump amplitude may also be modulated by an externalfunction generator, or by some other external means such as a chopperwheel fixtured in the pump beam path, with a reference signal suppliedto the lock-in amplifier. The pump and probe beams (from the pump laser103 and probe laser 104, respectively) are made collinear through theuse of the dichroic beamsplitter 105 and co-focused to a point along thebeam path using a microscope objective lens 106. The focal positions ofthe pump and probe beams are controlled with independent telescopinglens arrangements (107 and 108) in either input beam path, which may beautomated. The collinear and co-focused pump and probe beams aredirected to the semiconductor sample 109 using the microscope objective106. The microscope objective 106 is mounted on a stage effective totranslate the microscope objective along the common beam path. Thestage, which may be motorized and/or controlled by thecomputer/controller 116, provides for control of the displacement of themicroscope objective 106 from the sample 109. As the pump radiationinteracts with the sample, the sample obtains a reflectance modulation,which results changes in amplitude of the reflected probe light. Thesemiconductor sample 109 reflects the incident probe beam back throughthe microscope objective 106. The polarization beamsplitter 110,operating in conjunction with a quarter wave plate 111, is used toswitch the reflected probe beam out of the incoming probe beam path. Theintensity of the pump and probe beams at focus may also be controlledvia neutral density filters fixtured in either input beam path. In orderto remove any residual pump light and/or any photo-luminescence signal,the reflected probe beam is projected through a color filter 112 and/oronto a dielectric mirror 113. (Likewise, it may be appreciated anyattendant photo-luminescence signal may be separated through the use ofsuitable dielectric mirrors and/or color filters and analyzed in accordwith techniques known in the art.) The reflected probe beam is collectedat the collection lens 114 and input to the photoreceiver 115, therebyintegrating over the radial profile of the beam. The resulting Zdependence of the modulated reflectance depends solely on the focalparameters and the complex carrier diffusion length. The photoreceiveroutput signal (voltage or current) is passed to the lock-in amplifier102, which measures the PMR signal. The PMR signal is the relativechange in the radially integrated reflected probe light intensity andconsists of a vector characterized by an amplitude and a phase. Themicroscope objective 106 is stepped through sequence of positions withrespect to the sample surface. At each position, the PMR signal ismeasured and transmitted to the computer/controller 116, which recordsthe PMR signal as function of Z. The computer 116 may also be used toperform a nonlinear regression analysis to determine an electronictransport property of the sample as described hereinafter.

In an exemplary embodiment, the pump and probe beams are directed atnormal incidence onto a sample. The distance between the common beamwaist and the sample surface is Z. The focal geometry of the incidentpump and probe beams at Z=0 is illustrated in FIG. 2. At focus, thelinear reflected pump and probe beam profiles (203 and 204,respectively) will coincide with their respective input beams. As noted,the pump will induce a reflectance modulation within a radiusω_(m)=(ω_(p) ²+L_(d) ²)^(1/2). As the pump beam is stepped throughfocus, the area of modulation varies according to the relation ω_(m)²(Z)=ω_(p) ²(Z)+L_(d) ². The Fresnel coefficient for the reflected probebeam includes the changes due to pump-induced energy transformationprocesses. The mirror-reflected probe beam amplitude may be expanded asa sum of Gaussian beams of decreasing waist. In particular, given adominant photovoltage effect according to Eq. (1), and retaining onlytwo terms in the expansion, the electric field of the reflected probelaser beam at the surface of the sample may be written (disregarding thecommon spatial phase):

E _(r) =E _(o)ω_(o)/ω(Z)×exp{−ρ²/ω²(Z)}×[r+∂r/∂n×(n ₂ +ik ₂)I _(p)ω_(m)²/ω_(m) ²(Z)×exp{−2ρ²/ω_(m) ²(Z)}],  (3)

where |E_(o)|² is the intensity of the probe beam at focus, ω_(o) is theprobe beam waist (i.e. ω(Z)=ω_(o)[1+(Z/z_(o))²]^(1/2), where z_(o)=πω²/λis the Raleigh range of the probe beam), ρ is the radial distance asmeasured from the probe beam axis, r is the complex reflectancecoefficient, n is the index of refraction of the sample, n₂ and k₂ areeffective nonlinear indices defined by the coefficients appearing in Eq.(1), I_(p) is the intensity of the pump beam at focus, and ω_(m)(Z) isthe radius of modulation as previously defined. The leading termcorresponds to the DC component of the reflected probe beam whereas thesecond term corresponds to its modulated component. The broadening ofthe pump source by carrier diffusion is explicit in the factors ω_(m)and ω_(m)(Z) appearing in the second term. The modulated component ofthe reflected probe beam is also a Gaussian beam with radius defined bythe incident probe beam radius and the radius of modulation. At eachvalue of Z the reflected AC probe beam component retains a Gaussian formwith a smaller effective waist. FIG. 2 also shows the reflected AC probebeam profile 205 for different values of L_(d). When L_(d)≅0, the waistof AC reflected probe beam is smaller than that of either the pump orthe probe beam. However, when L_(d) ²>>ω_(p) ² the AC reflected beam 205will approach the reflected DC profile. Thus the profile of the ACreflected probe beam is dependent upon the diffusion length. In the caseof a dominant Drude effect, the expression for the mirror-reflectedprobe beam field at the surface is of precisely the same form as Eq.(3), the only distinction being that the effective nonlinear indices n₂and k₂ are now defined by the coefficients appearing in Eq. (2).

Squaring the mirror-reflected probe field and integrating the over thebeam profile yields the spatially integrated PMR signal via theidentity:

1+ΔR/R=[∫ ₀ ^(∞) |E _(r)|² ρdρ]/[∫ ₀ ^(∞) |E _(dc)|² ρdρ],  (4)

where E_(dc) is just the linear reflectance amplitude. Neglecting termsof second order in the nonlinear indices and performing the spatialintegrations in Eq. (4), the PMR signal may be written:

ΔR/R=4n ₂ I _(p)/(n ²−1)×(ω_(p) ² +L _(d) ²)/(ω²(Z)+ω_(p) ²(Z)+L _(d)²),  (5)

where n²>>k². The key observation here is that the Z dependence of thePMR signal is contained entirely within the denominator of Eq. (5).Moreover, if ω_(o) ²+ω_(p) ²≤L_(d) ², the 3D limit will be approachedfor Z=0 [J. Opsal et al., “Temporal behavior of modulated opticalreflectance in silicon,” J. Appl. Phys. 61, 240-248 (1987).]. However,well away from Z=0 (i.e. where ω²(Z)+ω_(p) ²(Z)≤L_(d) ²), the 1D limitis restored. (Note Eq. (3) is valid in the 3D limit. Thus the treatmentby the method of Gaussian decomposition smoothly interpolates betweenthe 3D and 1D limits.) Accordingly, the appearance of L_(d) ² in thedenominator of Eq. (5) shows the Z dependence of the PMR signal willdepend strongly on diffusion length provided the pump and probe beamwaists are commensurate with L_(d).

Furthermore, at “intermediate” modulation frequencies where therecombination lifetime is comparable to the modulation period (i.e.Ωτ˜1, where Ω is the modulation frequency in radians per second and τ isthe recombination lifetime), τ likewise becomes coupled into the Zdependence of the PMR signal through the appearance of the complexdiffusion length L_(d)→L_(d)/(1+iΩτ)^(1/2). In particular, Eq. (5)demonstrates that the PMR signal as a function of Z may be parameterizedby the expression:

ΔR/R=Aexp{iϕ _(o)}/[ω²(Z)+ω_(p) ²(Z)+L _(d) ²/(1+iΩτ)],  (6)

where A and ϕ_(o) are the Z independent PMR amplitude and phase,respectively. Remarkably, the dependencies of the PMR signal on variousparameters such as pump absorption depth and surface or interfacerecombination velocities are absorbed into A and ϕ_(o). Thus the Zdependence of the PMR signal is independent of the various parameterswhich determine the PMR amplitude and phase at Z=0. Thus the methodsdisclosed here do not require prior knowledge of such parameters or howthey enter into the PMR signal.

The coupling of the complex diffusion length into the Z dependence ofthe PMR signal indicates that L_(d), and ultimately τ, may be determinedby a regressive fit to the experimental Z-scan PMR data. For example,according to Eq. (6), the Z dependence of the PMR amplitude may beparameterized by the expression:

ΔR/R=A/[L _(d) ⁴+2L _(d) ²{ω²(Z)+ω_(p) ²(Z)}+{ω²(Z)+ω_(p)²(Z)}²(1+Ω²τ²)]^(1/2),  (7)

whereas the Z dependence of the PMR phase may be parameterized by theexpression:

ϕ=ϕ_(o)+tan⁻¹ {L _(d) ² Ωτ/[L _(d) ²+{ω²(Z)+ω_(p) ²(Z)}(1+Ω²τ²)]}.  (8)

Thus the Z dependence of the PMR amplitude may be parameterized by theset of variables: A, L_(d) ², ω_(p) ², ω_(o) ², and Ωτ, whereas thecorresponding phase expression allows analytic parametrization using thevariables: ϕ_(o), L_(d) ², ω_(p) ², ω_(o) ², and Ωτr. (Note that Ω,ω_(o), and ω_(o) are system parameters, whereas A, ϕ_(o), L_(d) ², and τare sample parameters. For analysis purposes, the system parameters maybe treated as fixed parameters, whereas the sample parameters may betreated as variable parameters to be resolved by the fit procedure.) Anonlinear regression analysis can then be used to adjust the variableswithin the appropriate parameterized expression to provide an optimumfit to the PMR phase and/or amplitude data. For example, as disclosed inU.S. Provisional Patent Application Ser. No. 62/498,685 (incorporatedherein by reference), the well-known Levenberg-Marquardt method may beused to adjust the variables within a nonlinear equation of the form ofEq. (7) in order to establish the carrier diffusion length and itsestimated uncertainty. (Note that in the limit Ωτ=0, Eq. (7) reduces to|ΔR/R≈A/[L_(d) ²+ω²(Z)+ω_(p) ²(Z)].)

It is to be appreciated the fit procedure is performed by a computerprogram, embodied on a non-transitory computer readable medium(including any medium that facilitates transfer of a computer programfrom one location to another), comprising executable code effective toreceive PMR amplitude and/or phase data acquired as a function of Z,system parameters (such as modulation frequency, focal parameters,etc.), and initial guesses for variable parameters, to perform thenonlinear regression analysis, and to output the best-fit parameters, astatistical estimate of the error in the output parameters, and anoverall “goodness-of-fit” measure, as necessary. Thus computer programresiding on a physically separated computer, a remote server, or thelike, may be used to perform the nonlinear regression analysis in accordwith the present invention, such use falling within the scope of thedisclosure.

At intermediate frequencies, we have a pair of nonlinear equations,containing a number of common variables, from which we attempt toextract two variables (i.e. L_(d) and τ). The parameters A, L_(d) ², andΩτ are generally correlated within the amplitude fit while theparameters ϕ_(o), L_(d) ², and Ωτ are generally correlated in the phasefit. However, the expression for the amplitude has strong dependenceupon L_(d), but only minimal dependence upon Ωτ. On the other hand, theexpression for the phase is strongly dependent upon Ωτ (i.e. via theargument of the arctangent appearing in Eq. (8)). Accordingly, thedecomposition of Eq. (6) into simultaneous equations for amplitude andphase allows an iterative fit procedure (with the estimateduncertainties output from a standard nonlinear least-squaresminimization procedure).

First, in order to provide an initial estimate of L_(d) ², aconventional nonlinear least-squares procedure may be used to adjust theparameters of Eq. (7), while holding Ωτ constant (˜1), to provide anoptimized fit to the Z-scanning PMR amplitude data. An exemplary FORTRANsubroutine which may be used to adjust the parameters of Eq. (7) whileholding Ωτ constant is as follows:

SUBROUTINE funcs(x,a,y,dyda,na) INTEGER na REAL x,y,a(na),dyda(na) REALPI, WT, LPU, LPR, DPU, DPR, WBD2, DENOM PI=4.0*ATAN(1.0) WT=0.87338LPU=.488 LPR=.375 DPU=x−a(5) DPR=x−a(6)WBD2=a(2)*(1.0+(LPU*DPU/PI/a(2))**2)WBD2=a(4)*(1.0+(LPR*DPR/PI/a(4))**2)+WBD2 DENOM=WBD2**2*(1.0+WT**2)DENOM=a(3)**2+2.*a(3)*WBD2+DENOM y=a(1)/SQRT(DENOM)+a(7)dyda(1)=1./SQRT(DENOM) dyda(3)=−a(1)/DENOM**1.5*(a(3)+WBD2)dyda(2)=−a(1)/DENOM**1.5*(a(3)+WBD2*(1.+WT**2))dyda(4)=dyda(2)*(1.0−(LPR*DPR/PI/a(4))**2)dyda(2)=dyda(2)*(1.0−(LPU*DPU/PI/a(2))**2)dyda(5)=2.*a(1)/DENOM**1.5*(a(3)+WBD2*(1.+WT**2))dyda(6)=dyda(5)*DPR/a(4)*(LPR/PI)**2dyda(5)=dyda(5)*DPU/a(2)*(LPU/PI)**2 dyda(7)=1.0 return ENDwhere x≡Z, y≡|ΔR/R|, a(1)≡A, a(2)≡ω_(p) ², a(3)≡L_(d) ², a(4)≡ω_(o) ²,a(5) and a(6) are the positions of pump and probe beam waists,respectively, and a(7) is a constant offset. Note here Ωτ (“WT”) is heldexplicitly constant. However, it should be appreciated a nonlinearequation of the form of Eq. (7) may include Ωτ as a variable fitparameter.

Next, holding L_(d) ² constant at its value as output from the amplitudefit, a conventional nonlinear least-squares procedure may be used toadjust the parameters of Eq. (8) to provide an optimized fit to theZ-scanning PMR phase data. Then the output value for Ωτ may be heldconstant in the amplitude fit in order to improve the estimate of L_(d)². This procedure can be iterated, holding each L_(d) ² output from theamplitude fit constant in the subsequent phase fit, and each Ωτ asoutput from the phase fit constant in the subsequent amplitude fit,until L_(d) ² and Ωτ approach limiting values. The amplitude and phasefits may also be used to provide estimated statistical uncertainties inL_(d) ² and Ωτ, respectively. The PMR measurement uncertainties approachthe ppm level whereas Eqs. (7) and (8) are broadly applicable.Therefore, in practice, the estimated uncertainties of the output fitparameters depend primarily upon the number and spacing of the datapoints in Z. Thus the polar form provides an advantageous means todetermine L_(d) and τ with high precision.

In order to demonstrate the Z-scanning laser PMR technique, a set ofsilicon samples with various p-type shallow junction structures wereevaluated. These samples were selected primarily because they wereprocessed using process conditions encountered in advanced silicondevice manufacturing. The large number of samples, all exhibiting largePMR signals, were also convenient to demonstrate the technique and itspracticality. The shallow junctions were formed in silicon (100)substrates by implantation of n-type dopant (As) followed by low-energyhigh-dose B implantation. Dopant activation was performed usingmillisecond timescale flash-lamp based annealing. A range of basetemperature and flash temperature targets were used to study dopantactivation, dopant diffusion, and material quality. The processconditions studied included: (i) flash target temperatures in the1250-1350° C. range, (ii) an additional thermal annealing of the Ascounter doped layer prior to B implantation, and (iii) use of a Geamorphizing implant (AI) to reduce B ion channeling. The AI processintroduces a layer of crystalline defects close to the sample surface.These defects reduce the carrier diffusion length and recombination timein the implanted region. SIMS data indicated post-activation B dopinglevels of ≈1×10¹⁹/cc at junction depths X_(j)=20 nm across the sampleset. The voltages across the ultra shallow junctions are relativelylarge, on the order of several volts, and may be estimated from theconventional 1D Poisson analysis (which also provides the electricfields). A sketch of the pump-on/pump-off carrier flows in the normaldirection is shown in FIG. 1 of Chism 2010. While these carrier flowsdramatically affect the values A and ϕ_(o), it is to be emphasized thedetermination of electronic transport properties from Z-scan PMRprofiles as disclosed here does not require prior knowledge of A orϕ_(o).

The Z-scanning laser PMR measurement system was configured with pump andprobe beams generated from the output of semiconductor diode lasers. Thepump beam generally comprises a laser output with at least one photonenergy greater than the smallest interband transition energy of asemiconductor material within the sample. Generally, the probe beam maycomprise a laser output with at least one photon energy effective todirectly detect the modulated photovoltage induced by the pump, or maycomprise a laser output with a least one photon energy effective todirectly detect the excess carrier density generated by the pump. Thepump and probe beam wavelengths were 488 and 375 nm, respectively. Thisprobe beam wavelength is near the lowest energy direct interbandtransition in Si, resulting in a dominant photovoltage effect. The 375nm probe light has an absorption depth in Si δ≈23 nm. Therefore, in thisembodiment, any detected photo-voltage necessarily occurs at or near thesurface (see e.g., Eq. (7) of Aspnes 1969). The pump and probe beamswere made collinear by use of a dichroic beamsplitter and wereco-focused to a micrometer scale spot on the surface of each sample. Asnoted, the pump and probe beam waists are selected to be commensuratewith the diffusion lengths to be measured. However, as may beappreciated, carrier diffusion lengths vary widely depending onsemiconductor materials, crystallinity, dopant concentrations, andsurface conditions. For example, the diffusion length of carriers incrystalline Si varies from ˜10⁻¹ cm to ˜10⁻⁵ cm as the doping densityvaries from ˜10¹⁵/cc to ˜10²⁰/cc. In situations where the diffusionlength is large the modulation frequency may be selected such that Ωτ≥1,which provides a means to maintain the diffusion length commensuratewith the pump and probe beam waists.

The pump laser output was directly modulated via a reference signal fromthe lock-in amplifier. In general, the phase in a modulated photovoltagemeasurement will exhibit an arctangent dependence approaching zero forΩτ≤1, and −π/4 for Ωτ≥1. Therefore, in the “intermediate” regime (Ωτ˜1),the phase will transition, almost linearly, from ≈0° to ≈90° (see e.g.,FIGS. 8, 13, 16 and/or 17 of Park 2001). For each of the wafers used inthis study, this linearity was confirmed over the frequency range600-900 kHz (see e.g., FIGS. 8-11 of U.S. Provisional Patent ApplicationSer. No. 62/498,685). Thus an optimal modulation period of 750 kHz wasselected.

In general, the modulation frequency is dictated by the carrierrecombination lifetime and the transport property to be measured. Forexample, the diffusion length may be accessed directly by setting themodulation frequency such that Ωτ≈0 (i.e. low frequency), such that theamplitude reduces to |ΔR/R|≈A/[L_(d) ²+ω²(Z)+ω_(p) ²(Z)]. Then we haveone nonlinear equation, containing a number of known system parameters,from which we may evaluate L_(d). Similarly, the diffusion coefficient Dmay be accessed directly by setting the modulation frequency such thatΩτ>>1, such that the recombination time is eliminated (i.e. for Ωτ>>1,the amplitude reduces to |ΔR/R|≈A/[(D/Ω)²+{ω²(Z)+ω_(p) ²(Z)}²]^(1/2)).Thus, at high frequencies, the amplitude equation may be used toevaluate D, or equivalently, the mobility (via the Einstein relationμ=qD/k_(B)T, where k_(B)T is the thermal voltage (≈26 meV)). Thecombination of high and low frequency Z-scan PMR data may also be usedto determine the recombination time via the relation τ=L_(d) ²/D, whereL_(d) and D are determined from low and high frequency analyses,respectively. Furthermore, in practice, the recombination lifetime maybe accessed directly by setting the modulation frequency such that Ωτ˜1(i.e. intermediate frequency), such that Eq. (8) may be used in anonlinear regression analysis in order to evaluate τ. An example FORTRANsubroutine which may be used in nonlinear least-squares minimization ofEq. (8) is as follows:

SUBROUTINE funcs(x,a,y,dyda,na) INTEGER na REAL x,y,a(na),dyda(na) REALPI, LPU, DPU, LPR, DPR, WBD2, DENOM PI=4.0*ATAN(1.0) LPU=.488 LPR=.375DPU=x−a(5) DPR=x−a(6) WBD2=a(2)*(1.0+(LPU*DPU/PI/a(2))**2)WBD2=a(4)*(1.0+(LPR*DPR/PI/a(4))**2)+WBD2 DENOM=a(3)+WBD2*(1.0+a(7)**2)y=ATAN(a(3)*a(7)/DENOM)+a(1) dyda(1)=1.dyda(3)=a(7)/(DENOM**2+(a(3)*a(7))**2)*WBD2*(1.0+a(7)**2)dyda(2)=−a(3)*a(7)/(DENOM**2+(a(3)*a(7))**2)*(1.0+a(7)**2)dyda(4)=dyda(2)*(1.0−(LPR*DPR/PI/a(4))**2)dyda(2)=dyda(2)*(1.0−(LPU*DPU/PI/a(2))**2) dyda(5)=2.*a(3)*a(7)dyda(5)=dyda(5)/(DENOM**2+(a(3)*a(7))**2)*(1.0+a(7)**2)dyda(6)=dyda(5)*DPR/a(4)*(LPR/PI)**2dyda(5)=dyda(5)*DPU/a(2)*(LPU/PI)**2dyda(7)=a(3)/(DENOM**2+(a(3)*a(7))**2)dyda(7)=dyda(7)*(a(3)+WBD2*(1.0−a(7)**2)) return ENDwhere x≡Z, y≡ϕ, a(1)≡ϕ_(o), a(2)≡ω_(p) ², a(3)≡L_(d) ², a(4)≡ω_(o) ²,a(5) and a(6) are the positions of pump and probe beam waists,respectively, and a(7)≡Ωτ.

As may be appreciated, carrier recombination lifetimes vary widelydepending on semiconductor materials, crystallinity, dopantconcentrations, and surface conditions. For example, the recombinationlifetime of carriers in crystalline Si varies from ˜2×10⁻⁴ s to ˜10⁻⁹ sas the doping density varies from ˜10¹⁵/cc to ˜10²⁰/cc. Maintaining thecriteria Ωτ˜1 over such a range of recombination lifetimes requiresmodulation frequencies ranging from ˜1 kHz to ˜160 MHz. These frequencyranges are accessible with wide bandwidth lock-in amplifiers such as theSignal Recovery model 7280 (operable from ½ Hz to 2 MHz), or theStanford Research models SR860 (operable from 1 mHz to 500 kHz), SR865A(operable from 1 mHz to 4 MHz), and SR844 (operable from 25 kHz to 200MHz). As can be appreciated, such wide bandwidth lock-in amplifiers mayprovide access to the low and high frequency ranges as well.

The pump and probe beam waists were overlapped via telescopingarrangements configured in the input arms of either beam. The entirereflected probe beam was collected and directed into a photoreceiver,thus radially integrating the beam. The photoreceiver was a high speedUV enhanced silicon photo-diode connected to a trans-impedance amplifiercircuit. As may be appreciated, it is advantageous to keep the probeintensity at a minimum, since any photo-injection of electron-hole pairsfrom the probe will necessarily offset the sample baseline condition(e.g. by reducing the latent field). Likewise, any CW component of thepump is undesirable. Excessive pump and probe beam intensities may beavoided via neutral density filters fixtured in the input arms of eitherbeam. For the exemplary samples used here the photo-injected carrierdensity was maintained in low-injection (ΔN/N_(e)<<1). However, if theprobe intensity is too low, detection may not be possible withconventional photodiodes. Thus photoreceiver embodiments include anavalanche photo-diode (APD) connected to an amplifier circuit such as,for example, the Hamamatsu model C12703 high-gain APD module configuredwith the Hamamatsu model S5344 short wavelength APD. Photoreceiverembodiments may also include a photomultiplier tube connected to anamplifier circuit. The photoreceiver output was passed to the lock-inamplifier, which measured the amplitude and phase of the reflectivitychange. This information was transmitted to the computer/controller,which records the components of the differential change in reflectivityvector as a function of Z. Thus Z-scanning PMR data was acquired on theexemplary samples.

Estimates for Ωτ were obtained via regressive fitting to the acquiredphase data. Then Ωτ was fixed in regressive fits to the acquiredamplitude data in order to yield L_(d) ². The estimated uncertainties inthe extracted parameters were also output from the fitting procedure.FIG. 3 shows experimental Z-scan PMR amplitude data and fits obtainedfrom samples with and without AI (302 and 303, respectively). Theamplitudes are symmetric with respect to Z, as anticipated. The PMRamplitude from the sample without AI 303 shows a relatively broad Zprofile, whereas the data from the sample with AI 302 exhibits anarrower profile. The more sharply peaked PMR amplitude as a function ofZ seen on the sample with AI evidences a shorter diffusion length. Thisbehavior was apparent in the amplitude data for all samples thatreceived the AI process, as expected. Likewise, FIG. 4 showsexperimental Z-scan PMR phase data and fits obtained from the same pairof samples as shown in FIGS. 3 (402 and 403, respectively). The phasesare again symmetric with respect to Z, in accord with Eq. (8). Theobserved phase lag of approximately 45° confirms operation in theintermediate frequency regime (see e.g., FIGS. 12-15 of U.S. ProvisionalPatent Application Ser. No. 62/498,685). The broader PMR phase as afunction of Z seen on the sample with AI 402 evidences a shorterrecombination lifetime. The more sharply peaked amplitude data 302corresponds to the broader phase data 402, demonstrating carrierrelaxation in the sample with AI happens more quickly and occurs over ashorter range than in the sample without AI. This behavior was apparentin the phase data for all samples that received the AI process, asexpected. The remarkable impact of near surface damage on the Z-scanprofile, as seen in FIGS. 3 and 4, demonstrates the embodiment discussedhere is primarily sensitive to carrier electronic parameters within theabsorption depth of the probe. (This near-surface specificity is a keyadvantage in semiconductor manufacturing.) The mobility and itsestimated uncertainty were obtained from the extracted parameters viathe Einstein relation. Table 1 lists fitted values of diffusion length,recombination time, and mobility for the subset of samples with AI,assuming a measurement uncertainty of 2 ppm for the PMR amplitude and0.13° for the PMR phase.

TABLE 1 Flash temp L_(d) [μm] τ [ns] μ [cm²/V · s] [° C.] Flash onlyPre-soak Flash only Pre-soak Flash only Pre-soak 1300/550 6.07 ± .026.01 ± .02  90.5 ± .5  87.6 ± .6 157 ± 2 159 ± 2 1300/550 (2X) 9.09 ±.03 8.63 ± .02  83.1 ± .4  67.0 ± .5 382 ± 4 427 ± 5 1300/600 9.36 ± .089.92 ± .08 159.9 ± .5 167.3 ± .5 211 ± 3 226 ± 4 1350/600 10.19 ± .05 11.68 ± .02  163.0 ± .5 167.9 ± .5 245 ± 4 312 ± 4

Systematic variations in extracted parameters with process conditionsare observed. The extracted carrier parameters show little sensitivityto the As thermal anneal (columns labeled “Pre-soak”). This is expectedsince the AI step occurred after the As thermal anneal (prior to Bimplantation). When the 1300° C./550° C. flash anneal is repeated, thediffusion length increases by a factor of ≅1.5×, while the recombinationlifetimes are reduced by ≈10%. When the base temperature of the flashanneal is increased to 600° C., the carrier recombination lifetimeroughly doubles, indicating this higher base temperature results inbetter removal of the AI damage. However, the observed diffusion lengthonly increases ≈10% (with respect to the repeated 1300° C./550° C.anneal). This behavior indicates the repeated 1300° C./550° C. flashanneal achieves good junction activation but does not completely annealthe AI damage. Thus, using the Z-scan PMR technique, the effect of theprocess conditions on carrier transport properties are easily observed.This capability allows a semiconductor device manufacturer to tailorprocess conditions to achieve (and control) the desired result, and istherefore of great practical value in semiconductor manufacturing. Forall samples tested, the measured mobilities comport with values expectedfrom the activated doping levels. The estimated uncertainties in theextracted mobilities remains less than 2% in all cases.

In sum, the analytic parameterization for the Z dependence of the PMRsignal in terms of carrier diffusion length enables the direct, highprecision determination of carrier diffusion lengths, recombinationlifetimes, and/or diffusion coefficients using a nonlinear regressivefit to data obtained from a simple optical arrangement. For example,once the diffusion length and recombination lifetime (and theirestimated uncertainties) are known from the fit procedure, these carriertransport properties may be used to evaluate the diffusion coefficient,or equivalently, the carrier mobility (via the Einstein relation).Alternatively, low and high frequency PMR amplitude curves can be fit toobtain the carrier diffusion length and diffusion coefficient,respectively, which then may be used to evaluate the recombinationlifetime. And once the mobility and the recombination time are known,the carrier effective mass may likewise be evaluated. In addition, thePMR amplitude at focus has been previously used to characterize activedoping concentration (i.e. through the dependence of Eq. (1) on N_(e))[Chism 2010]. Therefore, provided the active dopant concentration isdetermined from the PMR amplitude at focus (or otherwise), the mobilityas measured from the Z-scanning PMR technique may be used tocharacterize the sheet resistance R_(s) via the relation R_(s)∝1/μN_(e).Thus, it is clear the PMR technique disclosed here may be of immediatepractical use in semiconductor device manufacturing.

As to a further discussion of the manner of usage and operation of thepresent invention, the same should be apparent from the abovedescription.

With respect to the above description then, it is to be realized thatthe optimum dimensional relationships for the parts of the invention, toinclude variations in size, materials, shape, form, function and mannerof operation, assembly and use, are deemed readily apparent to oneskilled in the art, and all equivalent relationships to thoseillustrated in the drawings and described in the specification areintended to be encompassed by the present invention.

Therefore, the foregoing is considered as illustrative only of theprinciples of the invention. Further, since numerous modifications andchanges will readily occur to those skilled in the art, it is notdesired to limit the invention to the exact construction and operationshown and described, and accordingly, all suitable modifications andequivalents may be resorted to, falling within the scope of thedisclosure.

The disclosures of all patents, patent applications, and publicationscited herein are hereby incorporated herein by reference in theirentirety, to the extent that they provide exemplary, procedural, orother details supplementary to those set forth herein.

What is claimed is:
 1. A method of determining at least one electronic transport property of a semiconducting sample, the method comprising the steps of: (a) directing an amplitude modulated pump laser beam onto an area of a surface of the sample, wherein the pump laser beam comprises at least one wavelength with energy greater than the smallest interband transition energy of a semiconductor material within the sample, thereby inducing time periodic changes in electronic charge density within the sample such that the reflectance of the sample obtains a time periodic modulation; (b) directing a second probe laser beam onto an area of the surface the sample, said area obtaining the time periodic modulated reflectance of step (a), wherein the probe laser beam comprises at least one wavelength suitable for detecting the induced changes in the reflectivity of the sample; (c) collecting and directing the probe light reflected from the sample into a photoreceiver which produces an electrical output in response thereto, said output comprising a photo-modulated reflectance signal; (d) measuring the photo-modulated reflectance signal using a phase-locked detection circuit; (e) performing a series of photo-modulated reflectance measurements of steps (a), (b), (c), and (d), with the surface of the sample at a plurality of distances from the focal plane of the pump laser beam; (f) performing a nonlinear regression analysis using the information collected in steps (a), (b), (c), (d) and (e) wherein the nonlinear regression analysis uses at least one electronic transport property of the sample as a variable parameter to determine at least one electronic transport property of the sample; and (g) reporting at least one determined electronic transport property of the sample.
 2. The method of claim 1, further comprising: performing a second nonlinear regression analysis using the information collected in steps (a), (b), (c), (d) and (e) to determine at least one electronic transport property of the sample, wherein the second nonlinear regression analysis uses at least one electronic transport property of the sample as a variable parameter and at least one electronic transport property of the sample as determined in step (f) as a fixed value.
 3. The method of claim 1, wherein an electronic transport property of the sample is selected from the group consisting of a carrier diffusion length, the square of a carrier diffusion length, a carrier recombination lifetime, the product of the modulation frequency and a carrier recombination lifetime, a carrier diffusion coefficient, a carrier mobility, a carrier effective mass, a sheet resistance, and a sheet conductance.
 4. The method of claim 1, wherein the pump beam waist is selected to be within an order of magnitude of a carrier diffusion length or less.
 5. The method of claim 1, wherein the modulation period is selected to be within an order of magnitude of a carrier recombination lifetime or greater.
 6. The method of claim 1, wherein the nonlinear regression analysis comprises Levenberg-Marquardt minimization.
 7. The method of claim 1, wherein the nonlinear regression analysis comprises a parametric model of the form: |ΔR/R|=A/[D ²+2D{ω ²(Z)+ω_(p) ²(Z)}+{ω²(Z)+ω_(p) ²(Z)}²(1+ω²)]^(1/2) +C, where |ΔR/R| represents photo-modulated reflectance amplitude data, Z is the longitudinal displacement of the sample surface from focus, ω(Z) is the probe beam radius at the sample surface, ω_(p)(Z) is the pump beam radius at the sample surface, A is a variable parameter which represents an amplitude, D is a variable parameter which represents the square of a carrier diffusion length, ψ is a variable parameter which represents the product of the modulation frequency and a carrier recombination lifetime, and C is a variable parameter which represents a constant.
 8. The method of claim 1, wherein the nonlinear regression analysis comprises a parametric model of the form: ϕ=tan⁻¹ {Dψ/[D+{ω ²(Z)+ω_(p) ²(Z)}(1+ψ²)]}+ϕ_(o), where ϕ represents photo-modulated reflectance phase data, Z is the longitudinal displacement of the sample surface from focus, ω(Z) is the probe beam radius at the sample surface, ω_(p)(Z) is the pump beam radius at the sample surface, D is a variable parameter which represents the square of a carrier diffusion length, ψ is a variable parameter which represents the product of the modulation frequency and a carrier recombination lifetime, and ϕ_(o) is a variable parameter which represents a constant.
 9. The method of claim 1, wherein the nonlinear regression analysis comprises a parametric model of the form: |ΔR/R|=A/[D+ω ²(Z)+ω_(p) ²(Z)]+C, where |ΔR/| represents photo-modulated reflectance amplitude data, Z is the longitudinal displacement of the sample surface from focus, ω(Z) is the probe beam radius at the sample surface, ω_(p)(Z) is the pump beam radius at the sample surface, A is a variable parameter which represents an amplitude, D is a variable parameter which represents the square of a carrier diffusion length, and C is a variable parameter which represents a constant.
 10. The method of claim 1, wherein the nonlinear regression analysis comprises a parametric model of the form: ϕ=tan⁻¹ {Dψ/[D+ω ²(Z)+ω_(p) ²(Z)]}+ϕ_(o), where ϕ represents photo-modulated reflectance phase data, Z is the longitudinal displacement of the sample surface from focus, ω(Z) is the probe beam radius at the sample surface, ω_(p)(Z) is the pump beam radius at the sample surface, D is a variable parameter which represents the square of a carrier diffusion length, ψ is a variable parameter which represents the product of the modulation frequency and a carrier recombination lifetime, and ϕ_(o) is a variable parameter which represents a constant.
 11. The method of claim 1, wherein the nonlinear regression analysis comprises a parametric model of the form: |ΔR/R|=A/[(D/Ω)²+{ω²(Z)+ω_(p) ²(Z)}²]^(1/2) +C, where |ΔR/R| represents photo-modulated reflectance amplitude data, Z is the longitudinal displacement of the sample surface from focus, ω(Z) is the probe beam radius at the sample surface, ω_(p)(Z) is the pump beam radius at the sample surface, Ω is the modulation frequency, A is a variable parameter which represents an amplitude, D is a variable parameter which represents a carrier diffusion coefficient, and C is a variable parameter which represents a constant.
 12. The method of claim 1, wherein the nonlinear regression analysis comprises a parametric model of the form: ϕ=tan⁻¹{(D/Ω)/[ω²(Z)+ω_(p) ²(Z)]}+ϕ_(o), where ϕ represents photo-modulated reflectance phase data, Z is the longitudinal displacement of the sample surface from focus, ω(Z) is the probe beam radius at the sample surface, ω_(p)(Z) is the pump beam radius at the sample surface, Ω is the modulation frequency, D is a variable parameter which represents a carrier diffusion coefficient, and ϕ₀ is a variable parameter which represents a constant.
 13. An apparatus for determining at least one electronic transport property of a semiconducting sample, the apparatus comprising the means for: (a) directing an amplitude modulated pump laser beam onto an area of a surface of the sample, wherein the pump laser beam comprises at least one wavelength with energy greater than the smallest interband transition energy of a semiconductor material within the sample; (b) directing a second probe laser beam onto an area common with the area of step (a), wherein the probe laser beam comprises at least one wavelength suitable for detecting the induced changes in the reflectivity of the sample; (c) collecting and directing the probe light reflected from the sample into a photoreceiver which produces an electrical output in response thereto, said output comprising a photo-modulated reflectance signal; (d) measuring the photo-modulated reflectance signal using a phase-locked detection circuit; (e) performing a series of photo-modulated reflectance measurements of steps (a), (b), (c), and (d), with the surface of sample at a plurality of distances from the focal plane of the pump laser beam; (f) performing a nonlinear regression analysis using the information collected in steps (a), (b), (c), (d), and (e) wherein the nonlinear regression analysis uses at least one electronic transport property of the sample as a variable parameter to determine at least one electronic transport property of the sample; and (g) reporting at least one determined electronic transport property of the sample.
 14. An apparatus for determining at least one electronic transport property of a semiconducting sample, the apparatus comprising: (a) a first laser source producing an amplitude modulated pump laser beam, wherein the pump laser beam comprises at least one wavelength suitable for inducing time periodic changes in electronic charge density within the sample; (b) a second laser source producing a continuous wave probe laser beam, wherein the probe laser beam comprises at least one wavelength suitable for detecting time periodic changes in the reflectivity of the sample induced by the amplitude modulated pump laser beam; (c) an optical system effective to direct either laser beam onto a common area on a sample surface, to translate the focal plane of the amplitude modulated pump laser beam with respect to the sample surface, and to collect and direct probe light reflected from the sample into a photoreceiver; (d) a photoreceiver effective to generate an electrical output in response to changes in the reflected probe light amplitude; (e) a phase-locked detection circuit effective to measure the photoreceiver output signals; (f) a computer/controller effective to record a series of photo-modulated reflectance measurements as a function of the distance between the sample surface and the focal plane of the amplitude modulated pump laser beam; and (g) a computer program, embodied on a non-transitory computer readable medium, comprising executable code to perform a nonlinear regression analysis using at least the recorded information to determine at least one electronic transport property of the sample and to report at least one determined electronic transport property of the sample.
 15. The apparatus of claim 14, wherein the amplitude modulated pump laser beam is directed onto the sample at normal incidence.
 16. The apparatus of claim 14, wherein the continuous wave probe laser beam is collinear with the amplitude modulated pump laser beam.
 17. The apparatus of claim 14, wherein the continuous wave probe laser beam is directed onto the sample at an oblique angle of incidence.
 18. The apparatus of claim 14, wherein the photoreceiver comprises an avalanche photodiode connected to an amplifier circuit.
 19. The apparatus of claim 14, wherein the phase-locked detection circuit comprises a wide bandwidth lock-in amplifier.
 20. The apparatus of claim 14, wherein the computer/controller is effective to control the focal position of the pump beam with respect to the sample surface. 